L(s) = 1 | + (−3.58 − 3.58i)2-s + (2.12 + 2.12i)3-s + 17.7i·4-s + (−10.0 − 4.93i)5-s − 15.2i·6-s + (−18.4 + 2.02i)7-s + (34.9 − 34.9i)8-s + 8.99i·9-s + (18.2 + 53.6i)10-s + 70.3·11-s + (−37.6 + 37.6i)12-s + (20.4 + 20.4i)13-s + (73.2 + 58.7i)14-s + (−10.8 − 31.7i)15-s − 108.·16-s + (−0.412 + 0.412i)17-s + ⋯ |
L(s) = 1 | + (−1.26 − 1.26i)2-s + (0.408 + 0.408i)3-s + 2.21i·4-s + (−0.897 − 0.441i)5-s − 1.03i·6-s + (−0.994 + 0.109i)7-s + (1.54 − 1.54i)8-s + 0.333i·9-s + (0.577 + 1.69i)10-s + 1.92·11-s + (−0.905 + 0.905i)12-s + (0.436 + 0.436i)13-s + (1.39 + 1.12i)14-s + (−0.185 − 0.546i)15-s − 1.69·16-s + (−0.00588 + 0.00588i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.726573 - 0.216794i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.726573 - 0.216794i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.12 - 2.12i)T \) |
| 5 | \( 1 + (10.0 + 4.93i)T \) |
| 7 | \( 1 + (18.4 - 2.02i)T \) |
good | 2 | \( 1 + (3.58 + 3.58i)T + 8iT^{2} \) |
| 11 | \( 1 - 70.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + (-20.4 - 20.4i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + (0.412 - 0.412i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 - 88.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-37.1 + 37.1i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 + 116. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 227. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-83.7 - 83.7i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 - 438. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-199. + 199. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (-72.8 + 72.8i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (141. - 141. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + 380.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 349. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + (-8.87 - 8.87i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 860.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-817. - 817. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 - 698. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (191. + 191. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 412.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (267. - 267. i)T - 9.12e5iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.63208847429341215876879547816, −11.86560519382794676068389454432, −11.09130721300806963873595505309, −9.655188258690225397519337780168, −9.185929609804585887163645474622, −8.271357268707102160890880890532, −6.85434570883042828519998076876, −4.07650874127353763961034612114, −3.19669804674474357080562238996, −1.10452707538827579861736212563,
0.832090686406592759770204365138, 3.63120098339001815680203841541, 6.07885580007396674836708377070, 6.93848385639901394752112874045, 7.69630985911539476828427595805, 8.954252182430806175113578577841, 9.577048335839592653769150223643, 10.99538541557572507126606777336, 12.25663359663486716029896709593, 13.85581094565635125596230866014